Data loading and cohort selection

Loading

afs_orig = read.csv("MGRB_GnomAD_SweGen_cancer_UKBB_AFs_outerjoined_ss-auto-gwas-snps_hcr.csv.xz", stringsAsFactors = FALSE, header = TRUE)
models_orig = read.csv("../data/manual_polygenic_scores.hcr_tag_rescued.csv", stringsAsFactors = FALSE, header = TRUE)

# Key allele frequencies by VID
afs_orig$vid = paste(afs_orig$chrom, afs_orig$pos, afs_orig$ref, afs_orig$alt, sep = ":")
afs_orig = afs_orig[,!(colnames(afs_orig) %in% c("chrom", "pos", "ref", "alt"))]

# Add gnomad AFs to the models for imputation of missing variants
temp.gnomad_af = (afs_orig$nAA_gnomad*2 + afs_orig$nRA_gnomad) / (2*(afs_orig$nAA_gnomad + afs_orig$nRA_gnomad + afs_orig$nRR_gnomad))
models_orig$aaf = temp.gnomad_af[match(models_orig$vid, afs_orig$vid)]

# Create a full UKBB cohort by combining the age-stratified numbers
afs_orig$nRR_ukbb = afs_orig$nRR_ukbb_0_55 + afs_orig$nRR_ukbb_55_60 + afs_orig$nRR_ukbb_60_65 + afs_orig$nRR_ukbb_65_70 + afs_orig$nRR_ukbb_70_75 + afs_orig$nRR_ukbb_75_inf
afs_orig$nRA_ukbb = afs_orig$nRA_ukbb_0_55 + afs_orig$nRA_ukbb_55_60 + afs_orig$nRA_ukbb_60_65 + afs_orig$nRA_ukbb_65_70 + afs_orig$nRA_ukbb_70_75 + afs_orig$nRA_ukbb_75_inf
afs_orig$nAA_ukbb = afs_orig$nAA_ukbb_0_55 + afs_orig$nAA_ukbb_55_60 + afs_orig$nAA_ukbb_60_65 + afs_orig$nAA_ukbb_65_70 + afs_orig$nAA_ukbb_70_75 + afs_orig$nAA_ukbb_75_inf
afs_orig$nmissing_ukbb = afs_orig$nmissing_ukbb_0_55 + afs_orig$nmissing_ukbb_55_60 + afs_orig$nmissing_ukbb_60_65 + afs_orig$nmissing_ukbb_65_70 + afs_orig$nmissing_ukbb_70_75 + afs_orig$nmissing_ukbb_75_inf

# Convert afs from wide to long format
library(reshape2)
afs_long = melt(afs_orig, id.vars = c("rsid", "negstrand", "vid"), value.name = "count")
afs_long$cohort = gsub("^n(RR|RA|AA|missing)_", "", afs_long$variable)
afs_long$variable = gsub("_.*", "", afs_long$variable)
afs = dcast(afs_long, vid + rsid + negstrand + cohort ~ variable, value.var = "count")
afs = afs[,c("vid", "rsid", "negstrand", "cohort", "nRR", "nRA", "nAA", "nmissing")]

# Exclude ASRB samples -- prelim examination suggests they are rather
# poor quality, and we are not interested in their PRS distributions
# anyway.  Also exclude the various MGRB filtration options, as they
# apply only to rare variants.  Exclude SweGen as we don't have a good
# HQ bed for it.
cohorts.sel = c("mgrborig", "gnomad", "ukbb")
cohorts.main = c("mgrborig", "gnomad", "ukbb")
afs = afs[afs$cohort %in% cohorts.sel,]

Model selection

Choose polygenic models with at least 10 loci, with the exception of ShortLifespan:Deelen:10.1093/hmg/ddu139 (only six loci passing filters). For cancers, choose polygenic models only for cancers with a positive control cohort. In the case of multiple models for the same disorder, choose the most recent original publication where possible (ie exclude “meta” signatures if a good original report is available).

# Excluded due to insufficient size:
    # "CancerOfBladder:Fritsche:10.1016/j.ajhg.2018.04.001",
    # "LymphoidLeukemiaAcute:Fritsche:10.1016/j.ajhg.2018.04.001",
    # "LymphoidLeukemiaChronic:Fritsche:10.1016/j.ajhg.2018.04.001",
    # "MalignantNeoplasmOfTestis:Fritsche:10.1016/j.ajhg.2018.04.001",
    # "NonHodgkinsLymphoma:Fritsche:10.1016/j.ajhg.2018.04.001",
    # "PancreaticCancer:Fritsche:10.1016/j.ajhg.2018.04.001",
    # "APOE_rs429358:NA:NA",
    # "ShortHealthspan:Zenin:10.1038/s42003-019-0290-0",
    # "Deelen2019_90_disc",
    # "Deelen2019_99_disc",
# Excluded because a better alternative was available
    # "CancerOfProstate:Fritsche:10.1016/j.ajhg.2018.04.001",
    # "ColorectalCancer:Fritsche:10.1016/j.ajhg.2018.04.001",
    # "MelanomasOfSkin:Fritsche:10.1016/j.ajhg.2018.04.001",
    # "BreastCancer:Li:10.1038/gim.2016.43",
    # "BreastCancerFemale:Fritsche:10.1016/j.ajhg.2018.04.001",
    # "ShortParentalLifespan:Pilling:10.18632/aging.101334",
# Excluded because of issues with population-specific alleles between UK and European popns
    # "BasalCellCarcinoma:Chahal:10.1038/ncomms12510",
    # "BasalCellCarcinoma:Fritsche:10.1016/j.ajhg.2018.04.001",
# Excluded as superseded by Timmers / Pilling studies:
    # "Deelen2014_85_rep",
    # "Deelen2014_85_disc",
    # "Deelen2014_90_rep",
    # "Deelen2014_90_disc",

models.sel = c(
    "AF:Lubitz:10.1161/CIRCULATIONAHA.116.024143",
    "DiastolicBP:Warren:10.1038/ng.3768",
    "EOCAD:Theriault:10.1161/circgen.117.001849",
    "PulsePressure:Warren:10.1038/ng.3768",
    "SystolicBP:Warren:10.1038/ng.3768",

    "AlzheimersDisease:Lambert:10.1038/ng.2802",
    "ShortHealthspan:Zenin:10.1038/s42003-019-0290-0",
    "ShortParentalLifespan:Timmers:10.7554/eLife.39856",

    "Height:Wood:10.1038/ng.3097",

    "BreastCancer:Michailidou:10.1038/nature24284",
    "ColorectalCancer:Schumacher:10.1038/ncomms8138",
    "Melanoma:Law:10.1038/ng.3373",
    "ProstateCancer:Hoffmann:10.1158/2159-8290.CD-15-0315"

)
models = models_orig[models_orig$id %in% models.sel,]

Variant filtering and imputation

# Drop variants with low genotyping rate in any cohort in which that
# variant was detected.  Use the threshold of 97% genotyping rate
library(plyr)
temp.lowgt = ddply(afs[!is.na(afs$nRR),], .(cohort), function(d) mean(d$nmissing / (d$nRR + d$nRA + d$nAA + d$nmissing) >= 0.03))
temp.lowgt
##     cohort           V1
## 1   gnomad 1.596488e-03
## 2 mgrborig 8.066142e-05
## 3     ukbb 5.066883e-02
# The UKBB samples have rather a lot of dropouts here: ~ 5% of loci
# have a gt rate under 97%.

temp.gt_rate = 1 - daply(afs[!is.na(afs$nRR),], .(vid), function(d) {
    d = d[d$cohort %in% cohorts.main,]
    max(d$nmissing / (d$nRR + d$nRA + d$nAA + d$nmissing))})
mean(temp.gt_rate < 0.97)
## [1] 0.04621999
afs = afs[afs$vid %in% names(temp.gt_rate)[temp.gt_rate >= 0.97],]
mean(models$vid %in% names(temp.gt_rate)[temp.gt_rate >= 0.97])
## [1] 0.9481481
# ~5.0% of model loci lost by this filter
models = models[models$vid %in% names(temp.gt_rate)[temp.gt_rate >= 0.97],]

# Create a set of AFs for variants that have VCF entries in every cohort.  
# Note that given the relatively small size of some cohorts, this tends to 
# preferentially exclude rare variants from consideration, and will probably 
# attenuate the GRS differences.
afs.nmissing_per_cohort = tapply(is.na(afs$nRR[afs$cohort %in% cohorts.main]), afs$vid[afs$cohort %in% cohorts.main], sum)
afs.nomissing = afs[!(afs$vid %in% names(afs.nmissing_per_cohort[afs.nmissing_per_cohort > 0])),]
nrow(afs.nomissing) / nrow(afs)
## [1] 0.8785374
mean(models$vid %in% afs.nomissing$vid)
## [1] 0.671274

Cohort AF comparisons

Here we examine all GWAS-reported loci that passed filtering.

Overall distribution

temp.afs = afs.nomissing
temp.afs$fA = (temp.afs$nAA*2 + temp.afs$nRA) / (2*(temp.afs$nAA + temp.afs$nRA + temp.afs$nRR))
temp.afs = acast(temp.afs, vid ~ cohort, value.var = "fA", fill = NA)
pairs(temp.afs[,cohorts.main], pch = ".")

library(corrplot)
## corrplot 0.84 loaded
corrplot.mixed(cor(temp.afs[,cohorts.main]), lower = "ellipse", upper = "number")

temp.overall_cohort.pvals = sapply(cohorts.main[1:(length(cohorts.main)-1)], function(cohort_1) {
    cohort_1_idx = which(cohorts.main == cohort_1)
    afs_1 = temp.afs[,cohort_1]
    sapply(cohorts.main[(cohort_1_idx+1):length(cohorts.main)], function(cohort_2) {
        afs_2 = temp.afs[,cohort_2]
        test = wilcox.test(afs_1 - afs_2)
        test$p.value
    })
})

temp.overall_cohort.pvals
## $mgrborig
##       gnomad         ukbb 
## 7.208864e-01 2.784435e-17 
## 
## $gnomad
##         ukbb 
## 6.889826e-05
p.adjust(unlist(temp.overall_cohort.pvals), "holm")
## mgrborig.gnomad   mgrborig.ukbb     gnomad.ukbb 
##    7.208864e-01    8.353305e-17    1.377965e-04
temp.af.gnomad = temp.afs[,"gnomad"]
temp.af.ukbb = temp.afs[,"ukbb"]
temp.af.mgrb = temp.afs[,"mgrborig"]
hist(temp.af.ukbb - temp.af.mgrb, breaks = c(-Inf, seq(-0.05, 0.05, 0.001), Inf), col = "grey", border = FALSE, xlim = c(-0.05, 0.05))

hist(temp.af.ukbb - temp.af.gnomad, breaks = c(-Inf, seq(-0.05, 0.05, 0.001), Inf), col = "grey", border = FALSE, xlim = c(-0.05, 0.05))

# Interestingly much higher AF diversity for gnomAD, despite it being a larger cohort.  Population effects?
t.test(temp.af.ukbb - temp.af.mgrb)
## 
##  One Sample t-test
## 
## data:  temp.af.ukbb - temp.af.mgrb
## t = -7.2744, df = 21047, p-value = 3.603e-13
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -0.0004276082 -0.0002460822
## sample estimates:
##     mean of x 
## -0.0003368452
t.test(temp.af.ukbb - temp.af.gnomad)
## 
##  One Sample t-test
## 
## data:  temp.af.ukbb - temp.af.gnomad
## t = -4.7358, df = 21047, p-value = 2.196e-06
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -0.0006723020 -0.0002786997
## sample estimates:
##     mean of x 
## -0.0004755008

There is a very slight but statistically significant skew in the AFs between UKBB and MGRB/gnomAD: UKBB has ever so slightly lower representation of the alt allele than both MGRB (0.0003360821 less) and gnomAD (0.0004744416 less). I suspect a subtle technical effect.

To address: is this of concern for the PRS calculations? This slight bias would be of concern if both:

  1. The PRS allele effect direction was consistently skewed either to or away from the reference allele, and
  2. The magnitude of sum(skew x beta) was significant relative to any inter-cohort PRS differences.

2 is difficult to address before we calculate the actual PRS distributions. 1 however we can test now.

pheno_alleles = read.csv("../data/phenotype_associated_alleles.all.csv", stringsAsFactors = FALSE)
pheno_alleles = pheno_alleles[pheno_alleles$vid %in% afs.nomissing$vid,]
# Keep only alleles with consistent effect on a phenotype class
pheno_alleles = ddply(pheno_alleles, .(class, vid), function(d) {
    if (nrow(d) > 1 && (all(d$direction == 1) || all(d$direction == -1)))
        d = d[1,,drop=FALSE]
    d
})
write.csv(pheno_alleles, "../data/phenotype_associated_alleles.filt.csv", quote = FALSE, row.names = FALSE)
pheno_alleles.tests = ddply(pheno_alleles, .(class), function(d) {
    af.gnomad = temp.afs[d$vid, "gnomad"]
    af.mgrb = temp.afs[d$vid, "mgrborig"]
    af.ukbb = temp.afs[d$vid, "ukbb"]

    daf.mgrb_gnomad = (af.mgrb - af.gnomad) * d$direction
    daf.mgrb_ukbb = (af.mgrb - af.ukbb) * d$direction

    n.mgrb_gt_gnomad.protective = sum(af.mgrb > af.gnomad & d$direction == -1)
    n.mgrb_gt_gnomad.deleterious = sum(af.mgrb > af.gnomad & d$direction == 1)
    n.gnomad_gt_mgrb.protective = sum(af.mgrb < af.gnomad & d$direction == -1)
    n.gnomad_gt_mgrb.deleterious = sum(af.mgrb < af.gnomad & d$direction == 1)

    ft.mgrb_gnomad = fisher.test(matrix(c(n.mgrb_gt_gnomad.protective, n.gnomad_gt_mgrb.protective, n.mgrb_gt_gnomad.deleterious, n.gnomad_gt_mgrb.deleterious), nrow = 2))

    n.mgrb_gt_ukbb.protective = sum(af.mgrb > af.ukbb & d$direction == -1)
    n.mgrb_gt_ukbb.deleterious = sum(af.mgrb > af.ukbb & d$direction == 1)
    n.ukbb_gt_mgrb.protective = sum(af.mgrb < af.ukbb & d$direction == -1)
    n.ukbb_gt_mgrb.deleterious = sum(af.mgrb < af.ukbb & d$direction == 1)
    ft.mgrb_ukbb = fisher.test(matrix(c(n.mgrb_gt_ukbb.protective, n.ukbb_gt_mgrb.protective, n.mgrb_gt_ukbb.deleterious, n.ukbb_gt_mgrb.deleterious), nrow = 2))

    data.frame(
        class = d$class[[1]], 
        ref_cohort = "mgrb", 
        test_cohort = c("gnomad", "ukbb"), 
        deltaaf_direction.median = c(median(daf.mgrb_gnomad), median(daf.mgrb_ukbb)),
        p.value.wilcox = c(wilcox.test(daf.mgrb_gnomad)$p.value, wilcox.test(daf.mgrb_ukbb)$p.value),
        or.fisher = c(ft.mgrb_gnomad$estimate, ft.mgrb_ukbb$estimate),
        cil.fisher = c(ft.mgrb_gnomad$conf.int[[1]], ft.mgrb_ukbb$conf.int[[1]]),
        ciu.fisher = c(ft.mgrb_gnomad$conf.int[[2]], ft.mgrb_ukbb$conf.int[[2]]),
        p.value.fisher = c(ft.mgrb_gnomad$p.value, ft.mgrb_ukbb$p.value))
})

pheno_alleles.tests$p.value.wilcox.holm = p.adjust(pheno_alleles.tests$p.value.wilcox, "holm")
pheno_alleles.tests$p.value.fisher.holm = p.adjust(pheno_alleles.tests$p.value.fisher, "holm")
pheno_alleles.tests
##            class ref_cohort test_cohort deltaaf_direction.median
## 1 anthropometric       mgrb      gnomad            -1.277496e-04
## 2 anthropometric       mgrb        ukbb            -1.653798e-05
## 3    behavioural       mgrb      gnomad             2.728057e-05
## 4    behavioural       mgrb        ukbb            -2.988239e-04
## 5       depleted       mgrb      gnomad            -3.750209e-03
## 6       depleted       mgrb        ukbb            -1.449596e-03
##   p.value.wilcox or.fisher cil.fisher ciu.fisher p.value.fisher
## 1   3.668051e-01 1.1007675  0.7340203   1.653162   6.931847e-01
## 2   8.792646e-01 1.0006054  0.6673769   1.499636   1.000000e+00
## 3   6.599423e-01 0.9837027  0.6985942   1.384911   9.333614e-01
## 4   3.107888e-01 1.1153097  0.7921750   1.570790   5.573214e-01
## 5   3.717134e-10 2.6607300  1.9391732   3.662378   2.638859e-10
## 6   3.436825e-08 2.1320342  1.5567164   2.927677   1.050530e-06
##   p.value.wilcox.holm p.value.fisher.holm
## 1        1.000000e+00        1.000000e+00
## 2        1.000000e+00        1.000000e+00
## 3        1.000000e+00        1.000000e+00
## 4        1.000000e+00        1.000000e+00
## 5        2.230280e-09        1.583315e-09
## 6        1.718413e-07        5.252648e-06

Save the allele freqs for a supp table

temp.nRR = acast(afs.nomissing, vid ~ cohort, value.var = "nRR", fill = NA)[,cohorts.main]
temp.nRA = acast(afs.nomissing, vid ~ cohort, value.var = "nRA", fill = NA)[,cohorts.main]
temp.nAA = acast(afs.nomissing, vid ~ cohort, value.var = "nAA", fill = NA)[,cohorts.main]
temp.nmissing = acast(afs.nomissing, vid ~ cohort, value.var = "nmissing", fill = NA)[,cohorts.main]
temp.AC = 2*temp.nAA + temp.nRA
temp.AN = 2*(temp.nRR + temp.nRA + temp.nAA)
temp = cbind(temp.AC, temp.AN)
colnames(temp)[1:ncol(temp.AC)] = paste("AC.", colnames(temp.AC), sep = "")
colnames(temp)[(ncol(temp.AC)+1):ncol(temp)] = paste("AN.", colnames(temp.AN), sep = "")
colnames(temp) = gsub("mgrborig", "mgrb", colnames(temp))

# Suppress generation of the file with UKBB data:
# write.csv(temp, file = "data/gwas_afs.csv", row.names = TRUE)

# We likely don't have permission to release UKBB AFs on such a broad scale, so
# generate a no UKBB set also:
write.csv(temp[,!grepl("ukbb", colnames(temp))], file = "gwas_afs_noukbb.csv", row.names = TRUE)

Individual loci

g.test = function(tbl)
{
    expected = outer(rowSums(tbl), colSums(tbl)) / sum(tbl)
    logoe = log(tbl/expected)
    logoe[tbl == 0] = 0
    stat = 2*sum(tbl*logoe)
    pchisq(stat, prod(dim(tbl)-1), lower.tail = FALSE)
}

temp.locus_cohort.pvals = ddply(afs.nomissing[afs.nomissing$vid %in% models$vid & afs.nomissing$cohort %in% cohorts.main,], .(vid), function(d) {
    nR = d$nRR*2 + d$nRA
    nA = d$nAA*2 + d$nRA
    g.test(cbind(nR, nA))
    # g.test(as.matrix(d[,c("nRR", "nRA", "nAA")]))
})
colnames(temp.locus_cohort.pvals)[2] = "p.value"
temp.locus_cohort.pvals$p.value.mtc = p.adjust(temp.locus_cohort.pvals$p.value, "BH")
# temp.locus_cohort.pvals$p.value.mtc is now calibrated for average false rejection rate 
# (ie calling a SNP population-associated when it in fact isn't).

mean(temp.locus_cohort.pvals$p.value.mtc < 0.01)
## [1] 0.4721724
temp.locus_cohort.maxdeltaaf = ddply(afs.nomissing[afs.nomissing$vid %in% models$vid & afs.nomissing$cohort %in% cohorts.main,], .(vid), function(d) {
    nR = d$nRR*2 + d$nRA
    nA = d$nAA*2 + d$nRA
    AAF = nA/(nA+nR)
    max(AAF) - min(AAF)
})
colnames(temp.locus_cohort.maxdeltaaf)[2] = "maxdeltaaaf"

temp.locus_cohort.maxdeltaaf[order(temp.locus_cohort.maxdeltaaf$maxdeltaaaf),]
##                   vid  maxdeltaaaf
## 635   20:41851935:G:A 4.804901e-05
## 175     11:244552:A:G 5.472387e-04
## 698   3:163838015:A:C 5.854864e-04
## 848   5:176517326:T:C 6.520533e-04
## 865    5:58337481:T:G 6.896540e-04
## 581   2:232268312:T:C 8.648204e-04
## 64    1:242034263:A:G 8.926576e-04
## 561    2:19942473:G:A 8.944422e-04
## 800    4:61995613:A:G 9.804085e-04
## 344   15:32993111:C:T 1.013646e-03
## 126   10:12943973:C:T 1.036166e-03
## 431   17:43216281:C:T 1.120399e-03
## 940    6:43711981:T:C 1.242711e-03
## 87     1:65010606:T:G 1.410640e-03
## 1094   9:22062134:G:T 1.487932e-03
## 960    6:81792063:G:T 1.508864e-03
## 369   15:72842705:G:A 1.511067e-03
## 734    3:62133492:G:A 1.517305e-03
## 771   4:147993702:A:G 1.550150e-03
## 50    1:215046892:G:A 1.631525e-03
## 46    1:203766331:A:G 1.651584e-03
## 792    4:39503196:A:G 1.678110e-03
## 816   5:108625324:C:A 1.746323e-03
## 1095   9:22102165:C:T 1.753972e-03
## 35     1:17308254:T:C 1.809376e-03
## 465   18:19450303:A:G 1.867013e-03
## 463     17:800593:T:C 1.957503e-03
## 666   21:45867411:G:A 1.968592e-03
## 291   13:55934157:A:G 1.984521e-03
## 1114   9:99203606:T:C 2.015219e-03
## 547   2:179786068:T:C 2.214285e-03
## 222  12:125441159:T:C 2.246113e-03
## 365   15:67455630:C:T 2.312045e-03
## 447   17:64783539:C:T 2.316572e-03
## 1082  9:118826916:G:A 2.361107e-03
## 385   15:98615560:C:T 2.370182e-03
## 716   3:190815978:A:G 2.376253e-03
## 18    1:151259043:C:T 2.419978e-03
## 967   7:107259721:T:C 2.424953e-03
## 935    6:36659932:C:T 2.450101e-03
## 688   3:136107549:G:A 2.505036e-03
## 14     1:14105298:G:A 2.546910e-03
## 887   6:118569679:T:G 2.552647e-03
## 1011  8:110115372:C:T 2.598390e-03
## 158   11:11563879:C:T 2.615930e-03
## 301   13:80618435:A:G 2.616690e-03
## 982   7:148629759:C:T 2.621543e-03
## 719    3:27416013:C:T 2.622335e-03
## 552   2:191227755:A:G 2.638522e-03
## 480   18:57323149:C:A 2.650551e-03
## 1023  8:128106880:A:C 2.808417e-03
## 37    1:177279412:G:A 2.814433e-03
## 808    4:82204091:A:G 2.816117e-03
## 953    6:75452066:T:C 2.865091e-03
## 506   19:42683964:C:T 2.865981e-03
## 818   5:113748571:C:T 2.886389e-03
## 639   20:50141264:T:C 2.900362e-03
## 900   6:149608874:G:A 2.955662e-03
## 1013  8:115698881:G:A 2.962404e-03
## 485   18:77222862:T:G 2.971526e-03
## 333   14:90636206:G:A 3.008568e-03
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## 575   2:218688596:T:G 1.759716e-02
## 85     1:59653742:A:G 1.770383e-02
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## 996    7:27976563:G:A 1.775104e-02
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## 981   7:145720467:A:G 1.778719e-02
## 167   11:14404825:C:T 1.786384e-02
## 359   15:62379971:A:G 1.788374e-02
## 413   16:89875710:A:C 1.791630e-02
## 765    4:12963574:G:A 1.793358e-02
## 32    1:170346164:C:T 1.793654e-02
## 735    3:63967900:A:G 1.794274e-02
## 178   11:35972672:G:A 1.807000e-02
## 850   5:179730843:A:G 1.809441e-02
## 980   7:144074929:G:A 1.809577e-02
## 898   6:142703877:G:A 1.810963e-02
## 216  12:116391296:T:C 1.811836e-02
## 354   15:54455757:T:C 1.812501e-02
## 1030  8:130723728:A:G 1.816356e-02
## 375   15:78816057:G:T 1.818666e-02
## 124  10:126383088:C:A 1.820138e-02
## 106  10:104604916:T:G 1.822156e-02
## 509   19:45415713:G:A 1.827054e-02
## 1102   9:86661654:C:T 1.827304e-02
## 10    1:118854698:G:A 1.829201e-02
## 963    6:84277037:C:A 1.829773e-02
## 632   20:40534202:G:A 1.833764e-02
## 130   10:27890831:A:G 1.839150e-02
## 557   2:198077877:A:G 1.843233e-02
## 516    19:7244233:A:C 1.844957e-02
## 1027  8:128424792:A:G 1.851813e-02
## 601    2:42462930:A:G 1.855264e-02
## 373   15:78686823:A:G 1.866659e-02
## 2      1:10566215:A:G 1.871633e-02
## 476   18:46657358:C:T 1.872113e-02
## 341  15:101632867:G:A 1.877734e-02
## 231   12:24784139:G:A 1.885831e-02
## 362   15:66867861:A:G 1.889948e-02
## 603    2:43463637:T:C 1.893412e-02
## 276   13:23371474:T:C 1.898917e-02
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## 446    17:6474096:C:T 1.899675e-02
## 54    1:218975475:G:A 1.900778e-02
## 248   12:58256714:T:G 1.905047e-02
## 435   17:47402807:C:T 1.906915e-02
## 615    2:68495002:G:A 1.907063e-02
## 331   14:75038472:T:C 1.907516e-02
## 205   11:85867875:A:G 1.910143e-02
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## 1090  9:138698196:G:A 1.940762e-02
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## 1042   8:19493786:T:C 1.970229e-02
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## 886   6:117868051:T:C 1.978912e-02
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## 1035  8:141729463:C:T 2.881498e-02
## 895   6:134173151:A:G 2.887869e-02
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## 324   14:65567515:G:T 2.937272e-02
## 885   6:117490664:T:C 2.937322e-02
## 571   2:217935116:T:C 2.953341e-02
## 668   22:19983213:C:T 2.954049e-02
## 108  10:105324774:C:T 2.955258e-02
## 296   13:73957681:A:G 2.955401e-02
## 532   2:145645775:G:A 2.980645e-02
## 857    5:33989518:C:T 2.987911e-02
## 769   4:145565826:C:T 3.046719e-02
## 645   20:60958269:T:C 3.073592e-02
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## 17    1:149938898:T:C 3.104603e-02
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## 793    4:41261500:T:C 3.129284e-02
## 424   17:36101156:T:C 3.134972e-02
## 582   2:232322779:G:A 3.146841e-02
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## 304   13:87001719:T:C 3.152685e-02
## 593    2:28635740:T:C 3.163465e-02
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## 526   2:136187345:C:T 3.169046e-02
## 905   6:152437016:T:C 3.183736e-02
## 925    6:28747902:T:C 3.196243e-02
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## 240    12:4942953:T:G 3.219421e-02
## 528   2:142746083:G:A 3.231244e-02
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## 1067  9:105754656:T:C 3.364954e-02
## 191   11:65583066:G:T 3.380290e-02
## 11    1:119491784:A:G 3.438420e-02
## 486   19:10803395:A:G 3.464492e-02
## 381   15:89851580:G:A 3.475654e-02
## 820   5:127696022:A:C 3.477973e-02
## 567   2:208526140:T:C 3.485917e-02
## 1046   8:27195121:T:C 3.499621e-02
## 76     1:38559352:A:G 3.500686e-02
## 909   6:160833664:C:T 3.506576e-02
## 845   5:172994624:G:A 3.535245e-02
## 320   14:55245149:A:G 3.542141e-02
## 906   6:156587831:T:C 3.559188e-02
## 883   6:116451442:A:G 3.602886e-02
## 223   12:13075379:G:A 3.609164e-02
## 706   3:183355405:A:G 3.614213e-02
## 725    3:46600520:A:G 3.646873e-02
## 747   4:106084778:C:T 3.651783e-02
## 21    1:154404406:A:C 3.670752e-02
## 748   4:106216205:T:C 3.713331e-02
## 724    3:41243742:T:G 3.730801e-02
## 1112   9:98265780:C:T 3.781516e-02
## 384   15:95312071:T:C 3.847165e-02
## 437   17:53209774:A:C 3.849144e-02
## 822     5:1297488:C:T 3.849828e-02
## 517   2:101576199:G:A 3.900736e-02
## 713   3:186004526:T:C 3.933196e-02
## 376   15:79121776:C:T 3.986323e-02
## 860     5:4012694:G:A 3.986729e-02
## 139    10:5886734:C:T 3.987780e-02
## 644   20:60920887:G:A 4.081819e-02
## 902   6:151942194:T:G 4.149024e-02
## 272   13:21570246:T:G 4.210229e-02
## 591    2:26924823:A:C 4.308281e-02
## 637   20:48316027:C:T 4.369981e-02
## 924    6:26200677:A:G 4.507931e-02
## 226   12:18040964:A:C 4.572318e-02
## 746   4:106061534:C:A 4.717647e-02
## 752   4:111586468:A:G 4.918761e-02
## 225   12:14520701:A:G 4.978408e-02
## 823   5:131585958:A:G 5.031346e-02
## 518    2:10178479:C:T 5.415964e-02
## 313   14:32981484:G:A 5.612800e-02
## 754   4:111706275:A:G 5.732161e-02
hist(temp.locus_cohort.maxdeltaaf$maxdeltaaaf)

# On the basis of this histogram, set a max delta aaf threshold of
# 4%.  Drop loci with a delta AAF greater than this.
mean(temp.locus_cohort.maxdeltaaf$maxdeltaaaf < 0.04)
## [1] 0.9874327
temp.sel_loci = temp.locus_cohort.maxdeltaaf$vid[temp.locus_cohort.maxdeltaaf$maxdeltaaaf < 0.04]
afs = afs[afs$vid %in% temp.sel_loci,]
models = models[models$vid %in% temp.sel_loci,]

Attempt to model the cohort AAF differences.

temp.afs = afs
temp.afs$fA = (temp.afs$nAA*2 + temp.afs$nRA) / (2*(temp.afs$nAA + temp.afs$nRA + temp.afs$nRR))
temp.afs = acast(temp.afs, vid ~ cohort, value.var = "fA", fill = NA)

temp.af.gnomad = temp.afs[,"gnomad"]
temp.af.ukbb = temp.afs[,"ukbb"]
temp.af.mgrb = temp.afs[,"mgrborig"]

logit = function(p) log(p) - log(1-p)

pairs(cbind(mgrb = logit(temp.af.mgrb), ukbb = logit(temp.af.ukbb), delta = logit(temp.af.ukbb) - logit(temp.af.mgrb)), pch = ".")

pairs(cbind(mgrb = temp.af.mgrb, ukbb = temp.af.ukbb, delta = temp.af.ukbb - temp.af.mgrb), pch = ".")

pairs(cbind(mgrb = temp.af.mgrb, ukbb = temp.af.ukbb, deltastd = (temp.af.ukbb - temp.af.mgrb)/sqrt(temp.af.ukbb*(1-temp.af.ukbb))), pch = ".")

pairs(cbind(mgrb = logit(temp.af.mgrb), ukbb = logit(temp.af.ukbb), deltastd = (temp.af.ukbb - temp.af.mgrb)/sqrt(temp.af.ukbb*(1-temp.af.ukbb))), pch = ".")

hist((temp.af.ukbb - temp.af.mgrb)/sqrt(temp.af.mgrb*(1-temp.af.mgrb)), breaks = 25, col = "grey")

pairs(cbind(mgrb = logit(temp.af.mgrb), gnomad = logit(temp.af.gnomad), delta = logit(temp.af.gnomad) - logit(temp.af.mgrb)), pch = ".")

pairs(cbind(mgrb = temp.af.mgrb, gnomad = temp.af.gnomad, delta = temp.af.gnomad - temp.af.mgrb), pch = ".")

pairs(cbind(mgrb = temp.af.mgrb, gnomad = temp.af.gnomad, deltastd = (temp.af.gnomad - temp.af.mgrb)/sqrt(temp.af.gnomad*(1-temp.af.gnomad))), pch = ".")

pairs(cbind(mgrb = logit(temp.af.mgrb), gnomad = logit(temp.af.gnomad), deltastd = (temp.af.gnomad - temp.af.mgrb)/sqrt(temp.af.gnomad*(1-temp.af.gnomad))), pch = ".")

hist((temp.af.gnomad - temp.af.mgrb)/sqrt(temp.af.mgrb*(1-temp.af.mgrb)), breaks = 25, col = "grey")

# Looks like a sd norm makes the allele frequency difference largely independent of AAF.
# Therefore we can characterise the distribution of this normalised frequency difference
# and sample from it without needing to condition on the allele frequency.
# With > 20k loci we may as well just bootstrap directly instead of fitting a distribution
# and sampling.
norm_delta_aaf.mgrb_ukbb = (temp.af.mgrb - temp.af.ukbb)/sqrt(temp.af.ukbb*(1-temp.af.ukbb))
norm_delta_aaf.mgrb_gnomad = (temp.af.mgrb - temp.af.gnomad)/sqrt(temp.af.gnomad*(1-temp.af.gnomad))

Compare PRS distributions between cohorts

PRS calculation

Perform a ‘stress test’ for the PRS as suggested by Greg. We wish to show here that the PRS differences between MGRB and UKBB, or MGRB and GnomAD, are unlikely to be simply due to drift. To do this, we simulate hypothetical MGRB cohorts derived from either UKBB or GnomAD, that are consistent with the drift-only hypothesis.

The model:

POPULATION:    UK  --drift & ethnicity-->  Australian  --selection-->  Australian Wellderly
               |                               |                                |
               V                               V                                V
SAMPLE:       UKBB                        Australian*                          MGRB

The ideal comparison is the Australian sample vs MGRB. However, this Australian sample is not available. We simulate it by derivation from the UKBB sample, based on the MGRB - UKBB allele frequencies, on the assumption that the distribution of MAF_MGRB - MAF_UKBB will be very close to the distribution of MAF_Aus - MAF_UKBB. This is reasonable if we suppose that relatively few loci are linked to depleted MGRB phenotypes.

Proceed as follows:

  1. Calculate normalised MGRB - UKBB AAF differences using observed AFs. As these AFs were sampled, they already incorporate sampling error, and we do not need to perform an additional sampling stage later. \(D_i = \frac{F_{MGRB}_i - F_{UKBB}_i}{\sqrt{(F_{UKBB}_i)*(1-F_{UKBB}_i)}}\)
  2. Calculate expected MGRB PRS scores using observed AFs.
  3. For each bootstrap iteration \(k\):
    1. Take UKBB PRS AFs.
    2. Sample hypothetical MGRB AFs as \[f_{MGRB}^{(k)}_i = f_{UKBB}_i + d^{(k)}_i*\sqrt{(f_{UKBB}_i)*(1-f_{UKBB}_i)}\] where \(d^{(k)} \~ d\).
    3. Calculate expected MGRB PRS scores
  4. Compare the permuted MGRB scores (reflective of the null of population drift-only differences) to the observed UKBB and MGRB scores.
stressboot.nboot = 100000
stressboot.cohorts = c("ukbb", "gnomad")

temp.afs.nomissing.modelonly = afs.nomissing[afs.nomissing$vid %in% models$vid,]
temp.afs.nomissing.modelonly$aaf = (temp.afs.nomissing.modelonly$nAA + 0.5*temp.afs.nomissing.modelonly$nRA)/(temp.afs.nomissing.modelonly$nAA + temp.afs.nomissing.modelonly$nRA + temp.afs.nomissing.modelonly$nRR)

stressboot.afmat.orig = acast(temp.afs.nomissing.modelonly, vid ~ cohort, value.var = "aaf")
stressboot.afmat.orig = stressboot.afmat.orig[,c("mgrborig", stressboot.cohorts)]
stressboot.models = acast(models, id ~ vid, value.var = "coef")
stressboot.models[is.na(stressboot.models)] = 0
stressboot.models = stressboot.models[,rownames(stressboot.afmat.orig)]
stopifnot(rownames(stressboot.afmat.orig) == colnames(stressboot.models))

stressboot.scores.orig = (2 * stressboot.models %*% stressboot.afmat.orig) / rowSums(stressboot.models != 0)
names(dimnames(stressboot.scores.orig)) = c("model", "cohort")

stressboot.normafdelta = list("aus_from_ukbb" = norm_delta_aaf.mgrb_ukbb, "aus_from_gnomad" = norm_delta_aaf.mgrb_gnomad)

stressboot.tasks = expand.grid(bootiter = 1:stressboot.nboot, deriv_cohort = paste("aus_from_", stressboot.cohorts, sep = ""))
stressboot.pb = progress_estimated(nrow(stressboot.tasks))

stressboot.scores.boot = aperm(daply(stressboot.tasks, .(deriv_cohort, bootiter), function(d) {
    update_progress(stressboot.pb)
    stopifnot(nrow(d) == 1)
    source_cohort = gsub("^aus_from_", "", d$deriv_cohort)
    bootiter = d$bootiter
    this.normdelta = stressboot.normafdelta[[d$deriv_cohort]]
    set.seed(314159+bootiter-1)   # Will --> linked normdelta samples between the cohorts

    this.delta = sample(this.normdelta, nrow(stressboot.afmat.orig), replace = TRUE)
    this.aus_sim_af = stressboot.afmat.orig[,source_cohort] + this.delta*sqrt(stressboot.afmat.orig[,source_cohort]*(1-stressboot.afmat.orig[,source_cohort]))
    this.aus_sim_af = pmax(0, pmin(1, this.aus_sim_af))

    (2 * stressboot.models %*% this.aus_sim_af) / rowSums(stressboot.models != 0)
}), c(3, 1, 2))
names(dimnames(stressboot.scores.boot))[1] = "model"
saveRDS(stressboot.afmat.orig, "stressboot_afmat_orig.rds")
saveRDS(stressboot.models, "stressboot_models.rds")
print(rowSums(stressboot.models != 0))
##          AF:Lubitz:10.1161/CIRCULATIONAHA.116.024143 
##                                                  390 
##            AlzheimersDisease:Lambert:10.1038/ng.2802 
##                                                   12 
##         BreastCancer:Michailidou:10.1038/nature24284 
##                                                   64 
##       ColorectalCancer:Schumacher:10.1038/ncomms8138 
##                                                   35 
##                   DiastolicBP:Warren:10.1038/ng.3768 
##                                                   28 
##           EOCAD:Theriault:10.1161/circgen.117.001849 
##                                                   93 
##                          Height:Wood:10.1038/ng.3097 
##                                                  408 
##                         Melanoma:Law:10.1038/ng.3373 
##                                                   11 
## ProstateCancer:Hoffmann:10.1158/2159-8290.CD-15-0315 
##                                                   17 
##                 PulsePressure:Warren:10.1038/ng.3768 
##                                                   21 
##      ShortHealthspan:Zenin:10.1038/s42003-019-0290-0 
##                                                    3 
##    ShortParentalLifespan:Timmers:10.7554/eLife.39856 
##                                                    7 
##                    SystolicBP:Warren:10.1038/ng.3768 
##                                                   14
print(dim(stressboot.scores.orig))
## [1] 13  3
# Convert to data frames for ggplot
stressboot.dfs = list(original = melt(stressboot.scores.orig[,stressboot.cohorts]), boot = melt(stressboot.scores.boot))
temp.mgrb_scores = stressboot.scores.orig[,"mgrborig"]
temp.ukbb_sd = apply(stressboot.scores.boot[,"aus_from_ukbb",], 1, sd)   # Get SDs from UKBB to normalise scores for plotting
stressboot.dfs$original$value.mgrb = temp.mgrb_scores[stressboot.dfs$original$model]
stressboot.dfs$boot$value.mgrb = temp.mgrb_scores[stressboot.dfs$boot$model]
stressboot.dfs$original$value.rel = stressboot.dfs$original$value - stressboot.dfs$original$value.mgrb
stressboot.dfs$boot$value.rel = stressboot.dfs$boot$value - stressboot.dfs$boot$value.mgrb
stressboot.dfs$summary = ddply(stressboot.dfs$boot, .(model, deriv_cohort), function(d) {
    mean.rel = mean(d$value.rel)
    ci.rel = quantile(d$value.rel, c(0.025, 0.975))
    nneg = sum(d$value.rel < 0)
    npos = sum(d$value.rel > 0)
    p.value = 2 * (min(nneg, npos) + 0.5) / (nrow(d)+1)
    c(p.value = p.value, mean.rel = mean.rel, lcl.rel = ci.rel[[1]], ucl.rel = ci.rel[[2]], nneg = nneg, npos = npos) })
stressboot.dfs$summary$p.value.holm = p.adjust(stressboot.dfs$summary$p.value, "holm")
stressboot.dfs$summary$p.value.BH = p.adjust(stressboot.dfs$summary$p.value, "BH")
stressboot.dfs$summary$mean.rel.norm = stressboot.dfs$summary$mean.rel / temp.ukbb_sd[stressboot.dfs$summary$model]
stressboot.dfs$summary$lcl.rel.norm = stressboot.dfs$summary$lcl.rel / temp.ukbb_sd[stressboot.dfs$summary$model]
stressboot.dfs$summary$ucl.rel.norm = stressboot.dfs$summary$ucl.rel / temp.ukbb_sd[stressboot.dfs$summary$model]
model deriv_cohort p.value mean.rel lcl.rel ucl.rel nneg npos p.value.holm p.value.BH mean.rel.norm lcl.rel.norm ucl.rel.norm
AF:Lubitz:10.1161/CIRCULATIONAHA.116.024143 aus_from_ukbb 0.0254697 0.0001539 0.0000186 0.0002889 1273 98727 0.4584554 0.0704853 2.2313031 0.2700467 4.187125
AF:Lubitz:10.1161/CIRCULATIONAHA.116.024143 aus_from_gnomad 0.0000100 0.0006023 0.0003467 0.0008590 0 100000 0.0002600 0.0001300 8.7295956 5.0252141 12.450588
AlzheimersDisease:Lambert:10.1038/ng.2802 aus_from_ukbb 0.0016300 0.0044430 0.0018189 0.0072322 81 99919 0.0342297 0.0061285 3.2416338 1.3270681 5.276685
AlzheimersDisease:Lambert:10.1038/ng.2802 aus_from_gnomad 0.4454055 0.0019348 -0.0029239 0.0068132 22270 77730 1.0000000 0.5514545 1.4116159 -2.1333268 4.971015
BreastCancer:Michailidou:10.1038/nature24284 aus_from_ukbb 0.1039290 0.0000973 -0.0000205 0.0002144 5196 94804 1.0000000 0.2014051 1.6277837 -0.3426078 3.584383
BreastCancer:Michailidou:10.1038/nature24284 aus_from_gnomad 0.1482285 0.0001631 -0.0000581 0.0003833 7411 92589 1.0000000 0.2408713 2.7279741 -0.9708937 6.409991
ColorectalCancer:Schumacher:10.1038/ncomms8138 aus_from_ukbb 0.1862281 0.0002565 -0.0001251 0.0006365 9311 90689 1.0000000 0.2848195 1.3171974 -0.6425888 3.268759
ColorectalCancer:Schumacher:10.1038/ncomms8138 aus_from_gnomad 0.8344217 0.0000775 -0.0006394 0.0007936 41721 58279 1.0000000 0.8865599 0.3977467 -3.2835466 4.075410
DiastolicBP:Warren:10.1038/ng.3768 aus_from_ukbb 0.0851491 0.0010263 -0.0001432 0.0021916 4257 95743 1.0000000 0.1844898 1.7247084 -0.2406805 3.682979
DiastolicBP:Warren:10.1038/ng.3768 aus_from_gnomad 0.0001100 0.0043229 0.0021199 0.0065184 5 99995 0.0026400 0.0009533 7.2644743 3.5624394 10.954015
EOCAD:Theriault:10.1161/circgen.117.001849 aus_from_ukbb 0.0005500 0.0003312 0.0001447 0.0005196 27 99973 0.0120999 0.0028600 3.4573950 1.5100011 5.424016
EOCAD:Theriault:10.1161/circgen.117.001849 aus_from_gnomad 0.0002900 0.0006627 0.0003096 0.0010150 14 99986 0.0066699 0.0018850 6.9179573 3.2322752 10.595112
Height:Wood:10.1038/ng.3097 aus_from_ukbb 0.1084489 0.0000344 -0.0000077 0.0000763 5422 94578 1.0000000 0.2014051 1.6026750 -0.3575338 3.560448
Height:Wood:10.1038/ng.3097 aus_from_gnomad 0.0777692 0.0000712 -0.0000080 0.0001504 3888 96112 1.0000000 0.1838182 3.3185787 -0.3745018 7.012111
Melanoma:Law:10.1038/ng.3373 aus_from_ukbb 0.8524615 0.0000875 -0.0008361 0.0010104 42623 57377 1.0000000 0.8865599 0.1858653 -1.7759509 2.146199
Melanoma:Law:10.1038/ng.3373 aus_from_gnomad 0.9529005 -0.0000534 -0.0017990 0.0017017 52355 47645 1.0000000 0.9529005 -0.1134544 -3.8212453 3.614608
ProstateCancer:Hoffmann:10.1158/2159-8290.CD-15-0315 aus_from_ukbb 0.0029300 0.0007735 0.0002657 0.0012748 146 99854 0.0556694 0.0095224 3.0000430 1.0305161 4.944542
ProstateCancer:Hoffmann:10.1158/2159-8290.CD-15-0315 aus_from_gnomad 0.1331687 0.0007227 -0.0002199 0.0016659 6658 93342 1.0000000 0.2308257 2.8030143 -0.8531019 6.461811
PulsePressure:Warren:10.1038/ng.3768 aus_from_ukbb 0.3382066 0.0007900 -0.0008370 0.0024038 16910 83090 1.0000000 0.4628091 0.9574243 -1.0144417 2.913353
PulsePressure:Warren:10.1038/ng.3768 aus_from_gnomad 0.6848232 0.0006364 -0.0024303 0.0037069 34241 65759 1.0000000 0.7880521 0.7712948 -2.9454698 4.492809
ShortHealthspan:Zenin:10.1038/s42003-019-0290-0 aus_from_ukbb 0.3228668 0.0004000 -0.0003925 0.0012052 16143 83857 1.0000000 0.4628091 0.9852262 -0.9667595 2.968370
ShortHealthspan:Zenin:10.1038/s42003-019-0290-0 aus_from_gnomad 0.6971230 -0.0002971 -0.0017708 0.0011913 65144 34856 1.0000000 0.7880521 -0.7317064 -4.3615137 2.934141
ShortParentalLifespan:Timmers:10.7554/eLife.39856 aus_from_ukbb 0.0000100 0.0129934 0.0075328 0.0185327 0 100000 0.0002600 0.0001300 4.6370240 2.6882574 6.613880
ShortParentalLifespan:Timmers:10.7554/eLife.39856 aus_from_gnomad 0.0271097 0.0112661 0.0011758 0.0214100 1355 98645 0.4608654 0.0704853 4.0205958 0.4196148 7.640704
SystolicBP:Warren:10.1038/ng.3768 aus_from_ukbb 0.0016500 0.0045264 0.0016972 0.0073337 82 99918 0.0342297 0.0061285 3.1430432 1.1785241 5.092330
SystolicBP:Warren:10.1038/ng.3768 aus_from_gnomad 0.4281457 0.0021495 -0.0031346 0.0074542 21407 78593 1.0000000 0.5514545 1.4925387 -2.1765863 5.176041
stressboot.dfs$summary$plot_cohort = c("aus_from_ukbb" = "UKBB", "aus_from_gnomad" = "gnomAD")[stressboot.dfs$summary$deriv_cohort]
library(ggplot2)
ggplot(stressboot.dfs$summary, aes(x = model, y = mean.rel.norm, ymin = lcl.rel.norm, ymax = ucl.rel.norm, col = plot_cohort, group = plot_cohort)) + 
    geom_point(position = position_dodge(width = 0.7), size = 2) + geom_errorbar(position = position_dodge(width = 0.7), width = 0.3, lwd = 1) + 
    geom_hline(yintercept = 0, lty = "dashed") + 
    labs(y = "Polygenic score relative to MGRB, normalised", x = "Polygenic model", col = "Comparison cohort") + theme_bw() + coord_flip()

Investigate basis of PRS differences

for (i in 1:nrow(stressboot.dfs$summary))
{
    try(
    dafplot(
        afs1 = afs.nomissing[afs.nomissing$cohort == "mgrborig",], 
        afs2 = afs.nomissing[afs.nomissing$cohort == gsub("^aus_from_", "", stressboot.dfs$summary$deriv_cohort[i]),], 
        model = models[models$id == stressboot.dfs$summary$model[i],],
        main = stressboot.dfs$summary$model[i],
        xlab = "",
        # xlab = expression(paste("PRS ", beta)),
        ylab = sprintf("MAF MGRB - %s", gsub("^aus_from_", "", stressboot.dfs$summary$deriv_cohort[i])),
        mar = c(9, 4, 4, 2)+0.1,
        sub = sprintf("\ndelta_mean=%.3e  (%.3e-%.3e)\np.raw=%.4f  p.holm=%.4f  p.bh=%.4f", stressboot.dfs$summary$mean.rel[i], stressboot.dfs$summary$lcl.rel[i], stressboot.dfs$summary$ucl.rel[i], stressboot.dfs$summary$p.value[i], stressboot.dfs$summary$p.value.holm[i], stressboot.dfs$summary$p.value.BH[i]))
    )
}

sessionInfo()
## R version 3.5.3 (2019-03-11)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 18362)
## 
## Matrix products: default
## 
## locale:
## [1] LC_COLLATE=English_Australia.1252  LC_CTYPE=English_Australia.1252   
## [3] LC_MONETARY=English_Australia.1252 LC_NUMERIC=C                      
## [5] LC_TIME=English_Australia.1252    
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] ggplot2_3.1.1          kableExtra_1.1.0       knitr_1.22            
##  [4] knitrProgressBar_1.1.0 corrplot_0.84          plyr_1.8.4            
##  [7] reshape2_1.4.3         rmarkdown_1.12         RevoUtils_11.0.3      
## [10] RevoUtilsMath_11.0.0  
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_1.0.1        highr_0.8         compiler_3.5.3   
##  [4] pillar_1.3.1      R.methodsS3_1.7.1 tools_3.5.3      
##  [7] digest_0.6.18     gtable_0.3.0      evaluate_0.13    
## [10] tibble_2.1.1      viridisLite_0.3.0 pkgconfig_2.0.2  
## [13] rlang_0.3.4       rstudioapi_0.10   yaml_2.2.0       
## [16] xfun_0.6          withr_2.1.2       dplyr_0.8.0.1    
## [19] stringr_1.4.0     httr_1.4.0        xml2_1.2.0       
## [22] hms_0.4.2         tidyselect_0.2.5  grid_3.5.3       
## [25] webshot_0.5.1     glue_1.3.1        R6_2.3.0         
## [28] purrr_0.3.2       readr_1.3.1       magrittr_1.5     
## [31] scales_1.0.0      htmltools_0.3.6   assertthat_0.2.1 
## [34] rvest_0.3.3       colorspace_1.4-1  labeling_0.3     
## [37] stringi_1.4.3     lazyeval_0.2.2    munsell_0.5.0    
## [40] crayon_1.3.4      R.oo_1.22.0